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arXiv · 2403.11013

Improved Algorithm and Bounds for Successive Projection

Jiashun Jin·Zheng Tracy Ke·Gabriel Moryoussef·Jiajun Tang·Jingming Wang
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Abstract

Given a $K$-vertex simplex in a $d$-dimensional space, suppose we measure $n$ points on the simplex with noise (hence, some of the observed points fall outside the simplex). Vertex hunting is the problem of estimating the $K$ vertices of the simplex. A popular vertex hunting algorithm is successive projection algorithm (SPA). However, SPA is observed to perform unsatisfactorily under strong noise or outliers. We propose pseudo-point SPA (pp-SPA). It uses a projection step and a denoise step to generate pseudo-points and feed them into SPA for vertex hunting. We derive error bounds for pp-SPA, leveraging on extreme value theory of (possibly) high-dimensional random vectors. The results suggest that pp-SPA has faster rates and better numerical performances than SPA. Our analysis includes an improved non-asymptotic bound for the original SPA, which is of independent interest.

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Jiashun Jin, Zheng Tracy Ke, Gabriel Moryoussef, Jiajun Tang, Jingming Wang. 2024-03-16. Improved Algorithm and Bounds for Successive Projection. https://arxiv.org/abs/2403.11013

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